\chapter{Concluding Remarks}

This chapter provides some concluding remarks. In addition, limitations
as well as a description of the ongoing research along with some
suggestions for future works are presented.

\section{Conclusions}

% Reliability prediction is a valuable procedure to anticipate system
% failures and then let organizations to be prepared to apply the most
% suitable actions to avoid and/or overcome such undesirable
% situations.

This work proposed a \gls{pso}+\gls{svm} methodology for solving
reliability prediction problems. \gls{pso}+\gls{svm}
combined with a validation set approach to guide the search for
appropriate \gls{svr} parameters' values was validated with examples
from literature concerning reliability prediction based on time series
data. The results showed that \gls{pso}+\gls{svm} can achieve outcomes
comparable to or better than the ones provided by other time series
prediction tools, such as \gls{nn} and \gls{arima}.

Moreover, \gls{pso}+\gls{svm} was applied to an example involving data
from oil production wells. The \gls{tbf} was predicted by considering
specific characteristics of the system, differently from the other
examples, which were all from literature and based on time series
data. The input variables were both numerical and categorical. The
latter were transformed into indicator variables, so as to be used by
the \gls{svr} algorithm. In addition, the numerical input variables as
well as the output variable were scaled in [1,2] in order to avoid
scale problems and then to get improved solutions. Although the
original database presented non-homogeneous data and some problems
related to data gathering, the proposed \gls{pso}+\gls{svm}
methodology was able to provide quite small error values.

\gls{pso} was adopted to tackle the \gls{svm} problem in the specific
context of reliability prediction. The implemented \gls{pso} involved
an empirical manner to avoid particles from exiting the feasible
search space in early iterations of the algorithm by initially setting
a small value for their maximum velocities. Also, both {\it lbest} and
{\it gbest} communication network among particles were incorporated.

For all examples, apart from 13 runs in Example 1, the \gls{pso} runs
converged before reaching the maximum number of iterations. This
reflects the ability of the \gls{pso} algorithm to find good solutions
in early steps of the algorithm. Also, for every example, the
validation \gls{nrmse}, which guided the search for the parameters
(fitness), did not present considerable differences along the 30 runs,
which can be observed from the related standard deviations.

Furthermore, a comparison between {\it lbest} and {\it gbest}
\gls{pso} models was performed. The results for the specific examples
considered in this dissertation indicated that the {\it lbest} was
faster and also provided test \gls{nrmse} values statistically
comparable to or better than the ones yielded from the {\it gbest}
approach.

Therefore, gi\-ven the obtained re\-sults, the coupling of \gls{pso}
with \gls{svm} is a promi\-sing me\-thodology to tackle reliability
prediction problems based on time series or on data related to
specific features of the system, for example, obtained by a condition
monitoring procedure.

\section{Limitations}

Although the implemented \gls{pso} linked with {\sf{LIBSVM}} is quite
general and can be easily adapted to other application domains, the
results achieved in this dissertation are limited to regression problems in
the specific context of reliability prediction from time series data sets or
from system features. Thus, they can not be generalized to other applications
without a previous investigation of the behavior of the proposed
methodology applied to the problem under consideration. 

Additionally, \gls{pso} is a heuristic search procedure based on
probabilities, then it does not guarantee the convergence to optimum
point. However, it is a useful tool for complex objective functions
that, for example, have no defined derivatives in their domains or
whose search for their optimum is very burdensome. It is rather a
mechanism to go in the ``right'' direction so as to obtain good
solutions. In many contexts these good solutions are indeed valuable.

Furthermore, the generalization capability in all provided examples was
assessed by the test \gls{nrmse}. This is an indication of such
ability but by no means guarantees that the corresponding trained
machine will have a good performance when predicting outputs from
inputs not in the training, validation or test sets. If some time after
the prediction of an outcome the real value could be observed, this
new observation may be incorporated in the data set. With this procedure, the
\gls{svm} can be periodically retrained and thus improve its
performance in predicting the phenomenon under analysis.

This work considered that failure times and
\gls{tbf} predictions were related to systems subject to a single failure mode.
The modeling of various failure modes can be tackled analytically by means of
a competitive risks framework \cite{cooke1996}, in which the different failure
modes represent risks that compete for leading the system to a failed state. To
handle
different failure modes with an \gls{svm} approach, it
would be necessary to associate an \gls{svm} to each one of them. In this way,
the estimate of the system's very next failure time or \gls{tbf} would be
determined by the most critical failure mode at the moment, that is, the one
with the smallest failure time or \gls{tbf} returned by the associated
\gls{svm}. % Thus, the handling of different failure modes considering
% \gls{svm} modeling would be an extension of this work.

\section{Ongoing Research}

The following items can be cited as topics of ongoing or future
research effort:

\begin{itemize}
\item The application of the proposed methodology in the context of
  fault diagnostic and prognostic. The former usually demands a
  \gls{svm} classification or a multi-classification task and the
  latter, often requires a \gls{svr} based on metrics obtained via
  condition monitoring.
\item The combination of \gls{pso}+\gls{svm} with \gls{smdp} so as to
  allow for the development of a more comprehensive tool to support
  maintenance decisions.
\item A further investigation of the multi-objective approach for
  \gls{svm}. 
\item The \gls{svm} training problem may involve huge matrices, which
  can render the optimization procedure burdensome. The
  paralellization of the \gls{svm} code can be an alternative to this
  issue, by dividing the entire problem in smaller ones to be solved
  in parallel.
\item The handling of different failure modes considering a
  \gls{pso}+\gls{svm} modeling.
\end{itemize}
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